MTW Exercise 25.5 b) - killing vectors

hi there,
In this Ex ( see attached snapshot ), point b), the poisson bracket equation is not so straightforward to obtain.
Please correct my Poisson Bracket expansion here :
The first one which is provided is simpler :
[ε,η] = εμδμηρ - ημδμερ = ζη

hey Mate,
Thanks for your reply. Well it seems I got confused here. I had based my second commutation on the first one. In my very first attempt I had indeed written your expression but had forgotten that the other variable was x not either of the killing vectors ε or η....
Damn that got me wrong all the way !
At the beginning I had started with the Jacobi identity but did not check out ...
Thanks very much !

Yeah what you wrote down is fine for a Lie bracket of vector fields on a smooth manifold but what you want is a Poisson bracket of scalar fields that are functions of phase space coordinates (cotangent bundle to the smooth manifold).

Did you get the final result by the way? Feel free to ask if you get stuck elsewhere.